nLab
weak homotopy equivalence

Idea

There are (at least) two notions of equivalence in homotopy theory. Weak homotopy equivalence is ‘observational’, i.e., $f:X\to Y$ is a w.h.e. if when we look at it through the observations that we can make of it using homotopy groups or even the fundamental infinity-groupoid, it looks to be an equivalence. In contrast, homotopy equivalence is more ‘constructive’; $f$ is a h.e. if there exists an inverse for it (up to homotopy, of course). Note that both of these notions are weaker than mere isomorphism of topological spaces (homeomorphism) and so can be considered examples of weak equivalences.

There are actually two related concepts here: whether two spaces are weakly homotopy equivalent and whether a map between spaces is a weak homotopy equivalence. The former is usually defined in terms of the latter.

Definition for topological spaces

A weak homotopy equivalence between topological spaces is a continuous map which induces isomorphisms on all homotopy groups. More precisely, $f:X\to Y$ is a weak homotopy equivalence if

1. $f$ induces an isomorphism $f_{*}:{\Pi }_0(X)\to {\Pi }_0(Y)$ of sets of path component?s, and
2. for any basepoint $x\in X$ and any $n\ge 1$, $f$ induces an isomorphism $f_{*}:{\pi }_n(X,x)\to {\pi }_n(Y,f(x))$.

If $X$ and $Y$ are path-connected, then (1) is trivial, and it suffices to require (2) for a single (arbitrary) $x$, but in general one must require it for at least one $x$ in each path component.

It is tempting to try to restate the definition as ”$f$ induces an isomorphism $f_{*}:{\pi }_n(X,x)\to {\pi }_n(Y,f(x))$ for all $x\in X$ and $n\ge 0$,” but this is not literally correct; such a definition would be vacuously satisfied whenever $X$ is empty, without regard to what $Y$ might be. If you really want to go this way, therefore, you still must add a clause for ${\Pi }_{-1}$ (the truth value that states whether a space is inhabited), so the definition is no shorter.

The existence of a weak homotopy equivalence from $X$ to $Y$ is a reflexive and transitive relation, but it is not symmetric. If $X$ and $Y$ are related under the equivalence relation generated by the existence of a weak homotopy equivalence from $X$ to $Y$, we call them weakly homotopy equivalent and say that they have the same weak homotopy type.

Since the the existence of a weak homotopy equivalence is reflexive and transitive but not symmetric, we can state this explicitly as follows: there exists a zigzag? of weak homotopy equivalences $X\leftarrow \to \leftarrow \dots \to Y$. This is equivalent to saying that $X$ and $Y$ become isomorphic in the homotopy category of topological spaces with the weak homotopy equivalences inverted.

The homotopy category of Top with respect to weak homotopy equivalences is Ho(Top)${}_{\mathrm{whe}}$.

Properties

Any homotopy equivalence is a weak homotopy equivalence. It requires a little bit of thought to prove this, because $f$ and its homotopy inverse $g$ need not preserve any chosen basepoint. But for any $x\in X$ and any $n\ge 1$, we have a diagram

$$\begin{array}{ccccc}{\pi }_n(X,x)& & \to & & {\pi }_n(X,g(f(x)))\\ & \searrow & & \nearrow & & \searrow \\ & & {\pi }_n(Y,f(x))& & \to & & {\pi }_n(Y,f(g(f(x))))\end{array}$$\array{\pi_n(X,x) & & \to & & \pi_n(X,g(f(x)))\\ & \searrow && \nearrow && \searrow\\ && \pi_n(Y,f(x)) && \to && \pi_n(Y,f(g(f(x))))}

in which the two horizontal maps are isomorphisms because $gf$ and $fg$ are homotopic to identities. Hence, by the two-out-of-six property? for isomorphisms, the diagonal maps are also all isomorphisms.

Conversely, any weak homotopy equivalence between m-cofibrant spaces (spaces that are homotopy equivalent to CW complexes) is a homotopy equivalence.

Weak homotopy equivalences are the weak equivalences in the classical “Quillen” model structure on topological spaces, and also in the “mixed” model structure.

Other types of weak homotopy equivalence

A map of simplicial sets is called a weak homotopy equivalence if its geometric realization is a weak homotopy equivalence of topological spaces, as above. (Since the geometric realization of any simpicial set is a CW complex, in this case its geometric realization is actually a homotopy equivalence.) These are the weak equivalences in the classical model structure on simplicial sets.

Likewise, a functor between small categories is sometimes said to be a weak homotopy equivalence if its nerve is a weak homotopy equivalence of simplicial sets. These are the weak equivalences in the Thomason model structure on categories? (not the folk model structure).

Similarly, one can define weak homotopy equivalences between any sort of object that has a geometric realization, such as a cubical set, a globular set, an n-category, an n-fold category, and so on.

Note that in some of these cases, such as as simplicial sets, symmetric sets, and probably cubical sets, there is also a notion of “homotopy equivalence” from which this notion needs to be distinguished. A simplicial homotopy equivalence, for instance, is a simplicial map $f:X\to Y$ with an inverse $g:Y\to X$ and simplicial homotopies $X\times {\Delta }^1\to X$ and $Y\times {\Delta }^1\to Y$ relating $fg$ and $gf$ to identities.

Is there any reason for calling these ‘weak’ homotopy equivalences rather than merely homotopy equivalences? —Toby

Mike: By “these” I assume you mean weak homotopy equivalences of simplicial sets, categories, etc. My answer is yes. One reason is that in some cases, such as as simplicial sets, symmetric sets, and probably cubical sets, there is also a notion of “homotopy equivalence” from which this notion needs to be distinguished. A simplicial homotopy equivalence, for instance, is a simplicial map $f:X\to Y$ with an inverse $g:Y\to X$ and simplicial homotopies $X\times {\Delta }^1\to X$ and $Y\times {\Delta }^1\to Y$ relating $fg$ and $gf$ to identities.

Toby: Interesting. I would have guessed that any weak homotopy equivalence could be strengthened to a homotopy equivalence in this sense, but maybe not.

Tim: I think the initial paragraph is somehow back to front from a philosophical point of view, as well as a historical one. Homotopy theory grew out of studying spaces up to homotopy equivalence or rather from studying paths in spaces (and integrating along them). This leads to some invariants such as homology and the fundamental group. Weak homotopy type (and it might be interesting to find out when this term was first used) is the result and then around the 1950s with the development of Whitehead’s approach (CW complexes etc.) the distinction became more interesting between the two concepts.

I like to think of ‘weak homotopy equivalence’ as being ‘observational’, i.e. $f$ is a w.h.e if when we look at it through the observations that we can make of it, it looks to be an ‘equivalence’. It is ‘top down’. ‘Homotopy equivalence’ is more ‘constructive’ and ‘bottom up’. The idea of simple homotopy theory takes this to a more extreme case, (which is related to Toby’s query and to the advent of K-theory).

With the constructive logical side of the nLab becoming important is there some point in looking at this ‘constructive’ homotopy theory as a counter balance to the model category approach which can tend to be very demanding on the set theory it calls on?

On a niggly point, the homotopy group of a space is only defined if the space is non-empty so one of the statements in this entry is pedantically a bit dodgy!

Toby: I would say that it has a homotopy group at every point, and this is true even if it is empty. You can only pretend that it has a homotopy group, period, if it's inhabited and path-connected.

Anyway, how do you like the introduction now? You could add a more extensive History section too, if you want.

Tim: It looks fine. I would add some more punctuation but I’m a punctuation fanatic!!!

With all these entries I suspect that in a few months time we will feel they need some tender loving care, a bit of Bonsai pruning!! For the moment lets get on to more interesting things.

Do you think some light treatment of simple homotopy theory might be useful,say at a historical level?